Global Gevrey Regularity for the Bénard Convection in a Porous Medium with Zero Darcy-Prandtl Number

Summary.

In this paper, we prove the existence and uniqueness for the threedimensional B{é}nard convection model in a porous medium with zero Darcy-Prandtl number using the Galerkin procedure. In addition, we show that the solutions to this problem are analytic in time with values in a Gevrey class regularity. We also prove that the solution of the standard Galerkin method converges exponentially fast, in the wave number, to the exact solution. This gives an analytical justification to the two-dimensional computational results of Graham, Steen, and Titi [{J. Nonlin. Sci.} 3 (1993), 153—167].